These 26 problems practice Common Core standards 8.G.A.1 and 8.G.A.3. Translate, reflect and rotate points and polygons on the coordinate plane, identify a motion from its image, and use the fact that rigid motions keep lengths, angles and parallel lines.
The last six are word problems and multi-step tasks. Work from easiest to hardest, and draw each figure on graph paper when it helps.
Problem 1: Translate a point
Translate the point \((3,-2)\) 4 units right and 5 units up. Write the image.
Problem 2: Reflect a point in the x-axis
Reflect \((5,-3)\) across the \(x\)-axis. Write the image.
Problem 3: Reflect a point in the y-axis
Reflect \((-4,6)\) across the \(y\)-axis. Write the image.
Problem 4: Rotate a point 180 degrees
Rotate \((2,-7)\) by \(180^{\circ}\) about the origin. Write the image.
Problem 5: Rotate a point 90 degrees
Rotate \((3,5)\) by \(90^{\circ}\) counterclockwise about the origin. Write the image.
Problem 6: Rotate a point 270 degrees
Rotate \((4,-1)\) by \(270^{\circ}\) counterclockwise about the origin. Write the image.
Problem 7: Translate a triangle
Translate triangle \(JKL\) using the rule \((x,y)\to(x+3,\,y-4)\). Write the coordinates of the image.
Problem 8: Reflect a triangle
Reflect triangle \(PQR\) across the \(y\)-axis. Write the coordinates of the image.
Problem 9: Reflect a trapezoid
Reflect trapezoid \(ABCD\) across the \(x\)-axis. Write the coordinates of the image.
Problem 10: Rotate a triangle 90 degrees
Rotate triangle \(DEF\) by \(90^{\circ}\) counterclockwise about the origin. Write the coordinates of the image.
Problem 11: Rotate a parallelogram
Rotate parallelogram \(WXYZ\) with vertices \(W(1,1)\), \(X(5,1)\), \(Y(6,3)\), \(Z(2,3)\) by \(180^{\circ}\) about the origin. Write the image.
Problem 12: Rotate a triangle 270 degrees
Rotate triangle \(GHI\) with vertices \(G(-3,1)\), \(H(-3,4)\), \(I(-1,1)\) by \(270^{\circ}\) counterclockwise about the origin. Write the image.
Problem 13: Describe a translation
Segment \(AB\) has endpoints \(A(-2,1)\) and \(B(0,4)\). Its image has endpoints \(A^{\prime}(3,-3)\) and \(B^{\prime}(5,0)\). Describe the translation and write its rule.
Problem 14: Identify the reflection
Triangle \(A^{\prime}B^{\prime}C^{\prime}\) is the image of triangle \(ABC\) after one reflection. Across which axis was it reflected? Give the rule.
Problem 15: Identify the rotation
Triangle \(A^{\prime}B^{\prime}C^{\prime}\) is the image of triangle \(ABC\) after a rotation about the origin. Decide whether the rotation is \(90^{\circ}\), \(180^{\circ}\) or \(270^{\circ}\) counterclockwise.
Problem 16: Write the rule
A motion sends \((2,5)\) to \((-2,5)\) and \((-3,7)\) to \((3,7)\). Name the motion and write its rule.
Problem 17: What stays the same
Triangle \(DEF\) has \(DE=6\text{ cm}\), \(EF=8\text{ cm}\) and \(\angle E=90^{\circ}\). It is reflected across the \(y\)-axis. Find \(D^{\prime}E^{\prime}\), \(E^{\prime}F^{\prime}\) and the measure of \(\angle E^{\prime}\).
Problem 18: Parallel lines
Line segment \(p\) joins \((1,1)\) and \((3,5)\). Segment \(q\) joins \((3,1)\) and \((5,5)\). Both are rotated \(180^{\circ}\) about the origin. Show that the images are parallel.
Problem 19: Translate, then reflect
Triangle \(ABC\) has vertices \(A(1,2)\), \(B(3,2)\), \(C(1,5)\). Translate it 2 units left and 3 units down, then reflect the result across the \(y\)-axis. Write the final image.
Problem 20: Rotate, then reflect
Triangle \(PQR\) has vertices \(P(1,1)\), \(Q(4,1)\), \(R(4,3)\). Rotate it \(90^{\circ}\) counterclockwise about the origin, then reflect the result across the \(x\)-axis. Write the final image.
Problem 21: Park map
On a park map, a bench is at \((-6,2)\), where each unit is 1 yard. The designers place a second bench by reflecting the first across the \(y\)-axis, which is the center path. Where is the second bench, and how far apart are the benches?
Problem 22: Game piece
In a board game on a coordinate grid, Maya’s piece is at \((3,2)\). A card makes it rotate \(90^{\circ}\) counterclockwise about the origin, and a second card does it again. Where is the piece, and which single rotation gives the same result?
Problem 23: Wallpaper stamp
A stamp design has a vertex at \((1,1)\). Each new stamp is a translation of the previous one by 4 units right. Where is that vertex on the 4th stamp?
Problem 24: Find the original point
Point \(Q^{\prime}\) is at \((-3,8)\) after a \(270^{\circ}\) counterclockwise rotation about the origin. Find the original point \(Q\).
Problem 25: Two reflections
Reflect the point \((3,2)\) across the \(x\)-axis, then reflect the result across the \(y\)-axis. Which single rotation about the origin has the same result?
Problem 26: Legs and the right angle
Right triangle \(ABC\) has \(A(1,1)\), \(B(5,1)\) and \(C(5,4)\). Rotate it \(90^{\circ}\) counterclockwise about the origin. Show that both legs keep their lengths and that the angle at \(B^{\prime}\) is still a right angle.
See every chapter on the Grade 8 math page.